To what extent is a large space of matrices not closed under the product?

Abstract

Let K denote a field. Given an arbitrary linear subspace V of M_n(K) of codimension lesser than n-1, a classical result states that V generates the K-algebra M_n(K). Here, we strengthen this in three ways: we show that M_n(K) is spanned by the products of the form AB with A and B in V; we prove that every matrix in M_n(K) can be decomposed into a product of matrices of V; finally, when V is a linear hyperplane of M_n(K) and n>2, we show that every matrix in M_n(K) is a product of two elements of V.